Position vector of a particle as a function of time is given by $\overrightarrow{\mathrm{R}}=(\mathrm{asin}\mathrm{\omega t})\widehat{\mathrm{i}}+(\mathrm{acos}\mathrm{\omega t})\widehat{\mathrm{j}}+\left(\mathrm{bsin}{\mathrm{\omega }}_0\mathrm{t}\right)\widehat{\mathrm{k}}$ 
The particle appears to be performing simple harmonic motion along z direction, to an observer moving in xy plane.

The distance travelled by the observer himself in the time interval he sees the particle completing one oscillation is S .Then

(a) The observer is moving in xy plane with his position vector changing with time as
$\overrightarrow{\mathrm{R}}=(\mathrm{asin}\mathrm{\omega t})\widehat{\mathrm{i}}+(\mathrm{acos}\mathrm{\omega t})\widehat{\mathrm{j}}$
This is a circle with radius a. Angular speed of the observer is $\mathrm{\omega }$ and his linear speed is v = a$\mathrm{\omega }$.
He finds that the particle oscillates along z direction with angular frequency = $\mathrm{\omega }$₀.
(b) Time period of rotation of the particle is
$\begin{array}{l}{\mathrm{T}}_0=\frac{2\mathrm{\pi }}{{\mathrm{\omega }}_0}\\ \therefore \mathrm{s}=\mathrm{v}.{\mathrm{T}}_0=2\mathrm{\pi a}\left(\frac{\mathrm{\omega }}{{\mathrm{\omega }}_0}\right)\end{array}$